Looking at some variations, Black can win the pawn endgame even if the White rook captures both passed pawns. Therefore, the only viable defense for White has to come from creating a passed pawn of his own. The direct Rg6 is not enough by one tempo, which suggests Rg1+ as the solution, forcing the Black king in front of his pawn: 1 Rg1+ Kb2 2 Rg6 Kc2 3 Rxh6 b2 4 Rb6 b1=Q 5 Rxb1 Kxb1 6 h6 and White wins. Nice one!

rateodoro, very interesting. I could not spot white's resource here without an engine. In fact, it took a depth of 14 ply for it to find white's win: Ke7. Definitely something I'll look into when I have the time.